Yes. There's a story mode with cutscenes and such. The 3ds game in addition to the Wii U story is also getting a new epilogue story and Linkle's story, which ties into the original story in some way that hasn't been revealed yet.
In addition to the story mode, there's the "Adventure" mode. You need to finish missions, sometimes with specific characters, to get items and unlock ways of progressing through an 8-bit style map, eventually getting to missions that unlock new weapons, costumes and characters. The actual battles take place in reused locations from the story mode though. The original HWs had maps based on Zelda 1, 2nd Quest and 8-bit versions of Twilight Princess and Majora's Mask's overworlds. It's not known yet which maps will be featured in the 3ds version of the Adventure mode. The characters still talk and such in this mode in order to make some missions work, but it has no real story.